Integrated Multi-Echelon Supply Chain Design with Inventories under Uncertainty: MINLP Models, Computational Strategies

Fengqi You, Ignacio E. Grossmann*

Department of Chemical Engineering, Carnegie Mellon University Pittsburgh, PA 15213, USA

November, 2008

Abstract We address the optimal design of a multi-echelon supply chain and the associated inventory

systems in the presence of uncertain customer demands. By using the guaranteed service

approach to model the multi-echelon stochastic inventory system, we develop an optimization

model for simultaneously optimizing the transportation, inventory and network structure of a

multi-echelon supply chain. We formulate this problem as an MINLP with a nonconvex

objective function including bilinear, trilinear and square root terms. By exploiting the

properties of the basic model, we reformulate the problem as a separable concave minimization

program. A spatial decomposition algorithm based on Lagrangean relaxation and piecewise

linear approximation is proposed to obtain near global optimal solutions with reasonable

computational expense. Examplers for industrial gas supply chains with up to 5 plants, 100

potential distribution centers and 200 customers are presented.

1. Introduction Due to the increasing pressure for remaining competitive in the global market place,

optimizing inventories across the supply chain has become a major challenge for the process

industries to reduce costs and to improve the customer service.1, 2 This challenge requires

integrating inventory management with supply chain network design, so that decisions on the

locations to stock the inventory and the associated amount of inventory in each stocking

location can be determined simultaneously for lower costs and higher customer service level.

However, the integration is usually nontrivial for multi-echelon supply chains and their

* To whom all correspondence should be addressed. E-mail: grossmann@cmu.edu

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associated inventory systems in the presence of uncertain customer demands.3

The multi-echelon inventory management problem has been extensively studied in the

past several decades. However, most of the inventory literature only considers the optimization

of inventory decisions without integrating them with supply chain design and planning

decisions. Simpson4 first studied the control of multi-echelon inventory system for a serial

supply chain. In that paper, Simpson proposes the guaranteed service approach to describe the

mechanics of the serial inventory system, in which each stage operates a base-stock policy in

the face of random but bounded demand. Simpson’s results showed that the optimal inventory

policy for the serial system is an “all or nothing” policy, i.e. each stage either has no safety

stock, or carries enough stocks to decouple the downstream stages form the upstream stages.

Almost at the same time, Clark and Scarf5 also studied the optimal control policy of the

multi-echelon inventory system with each stage operating with a base-stock policy. In contrast

to Simpson’s work, Clark and Scarf consider that the uncertain demand at each stage to be

unbounded, and thus the service level and replenishment lead time of each stage depend on the

upstream adjacent supply stages’ stock level. To solve the problem, they proposed a dynamic

programming approach based on the calculation of recursions for stage inventory and

replenishment amount. Later, Federgruen and Zipkin6 extended the result of Clark and Scarf5 to

infinite horizon, and showed that a stationary order-up-to level echelon policy is optimal. Some

recent extensions of the work by Clark and Scarf5 include the work by Lee and Billington,7

which presents a supply chain operation model with multi-echelon inventory system operating

under a periodic-review base-stock system at Hewlett Packard, the work by Glasserman and

Tayur,8 which considers capacitated limits in the multi-echelon inventory model, and the work

by Ettl et al.,9 which analyzed the relationship between stochastic lead time and the inventory

level for a multi-echelon system.

Meanwhile, the guaranteed service approach, which is based on the idea of maximum

service time allowance in each inventory location, has drawn more attention due to its ease for

computation for large scale inventory systems. Graves10 proposed that the problem by

Simpson4 could be solved as a dynamic program. Later, different extensions of Simpson’s work

for assembly networks, distribution networks and spanning trees were given by Inderfurth,11, 12

Inderfurth and Minner,13 Graves and Willems,14 all of whom use dynamic programming to

solve the inventory optimization problem. Recently, Minner15 used a similar approach to

investigate the safety stock placement problem for reverse supply chains. Graves and

Willems16, 17 extended the guaranteed service model for general supply chain configuration

problem with safety stock optimization. Further extensions of the guaranteed service approach

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are given by Humair and Willems18 that accounts for supply chain network with clusters of

commonality, and Bossert and Willems19 that allows the multi-echelon inventory system to be

reviewed and replenished for an arbitrary review period at each stage. Inventory placement

problem for general acyclic supply chain network has recently been addressed by Magnanti et

al.,20 who present an efficient mathematical programming approach to solve the guaranteed

service multi-echelon inventory model instead of using the traditional dynamic programming

approach. Jung et al.21 considered the safety stock optimization for a chemical supply chain by

using a simulation-optimization framework, but the supply chain design decisions are not

jointly optimized.

Research on integrated supply chain network design and inventory management is

relatively new. Most of the literature in this area tends to approximate the inventory cost

coarsely without detailed inventory management policy,21-25 and most of it does not optimize

the safety stocks, but just considers the safety stock level as a given parameter that is treated as

the lower bound of the total inventory level,22 or the safety stock level is considered as the

inventory targets that would lead to some penalty costs if violated.23 There are, however,

several related works that offer relevant insights. Erlebacher and Meller24 developed a

nonlinear integer programming model for the joint location-inventory model. To solve the

highly nonlinear nonconvex problem efficiently, they use a continuous approximation and a

number of other heuristic methods. Daskin et al.25 and Shen et al.26 present a joint

location-inventory model, which extends the classical uncapacitated facility location model to

include nonlinear working inventory and safety stock costs for a two-stage network, so that

decisions on the installation of distribution centers (DCs) and the detailed inventory

replenishment decisions are jointly optimized. To simplify the problem, inventories in the

retailers are neglected, and they also assume that all the DCs have the same constant

replenishment lead time, and the demand at each customer has the same variance-to-mean ratio.

With the same assumptions, Ozsen et al.27 have extended the model to consider capacitated

limits in the DCs. Their work is further extended by Ozsen et al.28 to compare with the cases

where customers restrict to single sourcing and the case which customers allow multi-sourcing.

Another extension is given by Sourirajan et al.,29 in which the assumption on identical

replenishment lead time is relaxed while the assumption on demand uncertainty is still

enforced. Recently, You and Grossmann30 proposed a mixed-integer nonlinear programming

(MINLP) approach to study a more general model based on the one developed by Daskin et

al.25 and Shen et al.,26 relaxing the assumption on the identical variance-to-mean ratio for

customer demands.

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The objective of this work is to develop optimization models and solution algorithms to

address the problem of joint multi-echelon supply chain network design and inventory

management. By using the guaranteed service approach to model the multi-echelon inventory

system,4, 11-19 we capture the stochastic nature of the problem and develop an equivalent

deterministic optimization model. The model determines the supply chain design decisions

such as the locations of distribution centers (DCs), assignments of customer demand zones to

DCs, assignments of DCs to plants, shipment levels from plants to the DCs and from DCs to

customers, and inventory decisions such as pipeline inventory and safety stock in each node of

the supply chain network. The model also captures risk-pooling effects31 by consolidating the

safety stock inventory of downstream nodes to the upstream nodes in the multi-echelon supply

chain. The model is first formulated as a mixed-integer nonlinear program (MINLP) with a

nonconvex objective function, and then reformulated as a separable concave minimization

program after exploiting the properties of the basic model. To solve the problem efficiently, a

tailored hierarchical decomposition algorithm based on Lagrangean relaxation and piece-wise

linear approximation is developed to obtain near global optimal solutions within 1% optimality

gap with modest CPU times. Several computational examples for industrial gases supply

chains and performance chemical supply chains are presented to illustrate the application of the

model and the performance of the proposed algorithm.

The outline of this paper is as follows. Some basic concepts of inventory management with

risk pooling and the guaranteed service model for multi-echelon inventory system are

presented in Section 2. The problem statement is given in Section 3. In Section 4, we introduce

the joint multi-echelon supply chain design and inventory management model. Two small

illustrative examples are given in Section 5. To solve the large scale problem, an efficient

decomposition algorithm based on Lagrangean relaxation, piecewise linear approximation and

model property is presented in Section 6. In Section 7 we present our computational results and

analysis. We conclude the paper with some conclusions and future directions in Section 8.

2. Multi-Echelon Inventory Model In this section, we briefly review some inventory management models that are related to

the problem addressed in this work. Detailed discussion about single stage and multi-echelon

inventory management models are given by Zipkin3 and Graves and Willem,16 respectively.

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2.1. Single Stage Inventory Model under Base-Stock Policy

There are many control policies for single stage inventory systems, such as base stock

policy, (s,S) policy, (r, Q) policy, etc.3 Among these policies, the periodic review base stock

policy is most widely used in the inventory control practice. The reason is based on two facts.

As shown by Federgruen and Zipkin,6 the base stock policy is optimal for single stage

inventory system facing stationary demand. For multi-echelon inventory system, the base stock

policy, though not necessarily optimal, has the advantage of being simple to implement and

close to the optimum.32 Before introducing the multi-echelon inventory model, we first review

the single stage base stock policy, which is the common building block for most of the

multi-echelon inventory models.

Figure 1 shows the inventory profile for a given product in a stocking facility operated

under the periodic review base stock policy. As can be seen, the inventory level decreases due

to the customer demand and increases when replenishments arrive. Under the periodic review

base stock policy, inventory is reviewed at the beginning of each review period and the amount

of the difference between a specified base stock level and the actual inventory position

(on-hand inventory plus in-process inventory minus backorders) is ordered for replenishment.

It is interesting to note that the well-known continuous review (r, Q) policy can be treated as a

special case of base stock policy with base stock level equal to r Q+ .

[Figure 1, (a), (b)]

Since under the base stock policy the lengths of the review period and the replenishment

lead time are determined exogenously, the only control variable is the base stock level. To

determine the optimal base stock level for a single stage inventory system, let us denote the

review period as p, the replenishment lead time as l and the average demand at each unit of time

is μ . Recall that the inventory position is the total material in the system (on-hand plus

on-order), and we start each review period with the same inventory position S, which is the

base stock level. We must wait p units of time to review the inventory position again and place

an order for replenishment, and then the order will take another l units of time to arrive (Figure

1). Therefore, the inventory in the system at the beginning of a review period should be large

enough to cover the demand over review period p plus the replenishment lead time l, i.e. the

optimal base stock level should be ( )p lμ + if demand is constant.

[Figure 2] However, under demand uncertainty the demand process is not constant and we need more

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inventory (safety stock) to hedge against stockout before we get a chance to reorder (Figure 2).

The acceptable practice in this field is to assume a normal distribution of the demand,

although of course other distribution functions can be specified. If the demand at each unit of

time is normally distributed with mean μ and standard deviation σ , the demand over review

period p and the replenishment lead time l is also normally distributed with mean ( )p lμ +

and standard deviation p lσ + . It is convenient to measure safety stock in terms of the

number of standard deviations of demand, denoted as safety stock factor, λ . Then the optimal

base stock level is given by,

( )S p l p lμ λσ= + + +

We should note that if α is the Type I service level (the probability that the total

inventory on hand is more than the demand) used to measure of service level, the safety stock

factor λ corresponds to the α -quantile of the standard normal distribution, i.e.

Pr( )x λ α≤ = .

2.2. Risk Pooling Effect

For single echelon inventory system with multiple stocking locations, Eppen31 proposed

the “risk pooling effect”, which states that significant safety stock cost can be saved by

grouping the demand of multiple stocking locations. In particular, Eppen considers a single

period problem with N retailers and one supplier. Each retailer i has uncorrelated normally

distributed demand with mean iμ and standard deviation iσ . The replenishment lead times

for all these retailers are the same and given as L and all the retailers guarantee the same Type

I service level with the same safety stock factor λ . Eppen compared two operational modes of

the N-retailer system: decentralized mode and centralized mode. In the decentralized mode,

each retailer orders independently to minimize its cost. Since in this mode the optimal safety

stock in retailer i corresponding to the safety stock factor λ is iLλ σ ,31 the total safety

stock in the system is given by,

1

Nii

Lλ σ=∑ .

In the centralized mode, all the retailers are considered as a whole and a single quantity is

ordered for replenishment, so as to minimize the total expected cost of the entire system. Since

in the centralized mode all the retailers are grouped, and the demand at each retailer follows a

normal distribution 2( , )i iN μ σ , the total uncertain demand of the entire system during the order

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lead time will also follow a normal distribution with mean 1

Nii

L μ=∑ and standard deviation

21

Nii

L σ=∑ . Therefore, the total safety stock of the distribution centers in the centralized

mode is given by,

21

Nii

Lλ σ=∑

which is less than 1

Nii

Lλ σ=∑ . Eppen’s simple model illustrates the potential saving in safety

stock costs due to risk pooling. For example, consider a single echelon inventory system with

100 retailers. Each retailer has uncorrelated normal distributed demand with identical mean μ

and standard deviation σ . Thus, the total safety stock of this system is 100z Lσ under

decentralized mode, and 10z Lσ under centralized mode, i.e. 90% safety stocks in this

system can be saved by risk pooling.

2.3. Guaranteed Service Model for Multi-Echelon Inventory

One of the most significant differences between single stage inventory system and

multi-echelon inventory system is the lead time. For single stage inventory system, lead time,

which may include material handling time and transportation time, is exogenous and generally

can be treated as a constant. However, for a multi-echelon inventory system, lead time of a

downstream node depends on the upstream node’s inventory level and demand uncertainty, and

thus the lead time and internal service level are stochastic. Based on this fact, simply

propagating the single stage inventory model to multi-echelon system will lead to a suboptimal

solution, since the optimization of a multi-echelon inventory system is usually nontrivial.3

There are two major approaches to model the multi-echelon inventory system, the

stochastic service approach and the guaranteed service approach.16 For a detailed comparison

of these two approaches, see Graves and Willems,16 and Humair and Willems.18 Briefly, the

stochastic service approach employs a more complicated model that allows for a more exact

and detailed understanding of the system. However, the model as well as the solution technique,

are not easy to develop and is computationally hard. The guaranteed service approach models

the entire system in an approximate fashion and allows a planner to make strategic and tactical

decisions without the need to approximate portions of the system that are not captured by a

simplified topological representation. Thus, the stochastic service model may be too

complicated to be employed for supply chain design problem. Based on this reason, we choose

the guaranteed service approach to model the multi-echelon inventory system in this work.

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The main idea of the guaranteed service approach is that each node j in the multi-echelon

inventory system quotes a guaranteed service time jT , by which this node will satisfy the

demands from its downstream customers. That is, the customer demand at time t must be

ready to be shipped by time jt T+ . For instance, if a chemical supplier’s guaranteed service

time is 3 days, it means that once the customer places an order, the total amount of chemicals

ordered by the customer will be available in “at most” 3 days. The guaranteed service times for

internal customers are decision variables to be optimized, while the guaranteed service time for

the nodes at the last echelon (facing external customers) is an exogenous input, presumably set

by the marketplace. Besides the guarantee service time, we consider that each node has a given

deterministic order processing time, jt , which is independent of the order size. The order

processing time, including material handling time, transportation time and review period,

represents the time from all the inputs that are available until the outputs are available to serve

the demand. Therefore, the replenishment lead time, which represents the time from when we

place an order to when all the goods are received, can be determined by the guaranteed service

time of direct predecessor 1−jT (the time that predecessor requires to have the chemicals ready

to be shipped) plus the order processing time jt (transportation time, handling time and

review period). As we can see from Figure 3, the net lead time of node j, jNLT , which is the

time span over which safety stock coverage against demand variations is necessary, is defined

as the difference between the replenishment lead time of this node and its guaranteed service

time to its successor.15 The reason is that not all the customer demand at time t for node j

should be satisfied immediately, but only needs to be ready by time jt T+ . Thus, the safety

stock does not need to cover demand variations over the whole replenishment lead time, but

just the difference between the replenishment lead time and the guaranteed service time to the

successors that is defined as the net lead time. Therefore, we can calculate the net lead time

with the following formula:

1−= + −j j j jNLT T t T

where node j-1 is the direct predecessor of node j (Figure 4). Note that this formula suggests

that if the service time quoted by node j to its successor nodes equals the replenishment lead

time 1− +j jT t , no safety stock is required in node j because all products are received from the

predecessors and processed within the guaranteed service time, i.e. this node is operating in

“pull” mode. If the guaranteed service time jT is 0, the node holds the most safety stock

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because all the orders, once they are placed, are fulfilled immediately, i.e. this node is operating

in “push” mode.

[Figure 3] [Figure 4]

In the guaranteed service approach, each node in the multi-echelon inventory system is

assumed to operate under a periodic review base stock policy with a common review period.

Furthermore, demand over any time interval is also assumed to be normally distributed (mean

jμ and standard deviation jσ for node j ), and “bounded”. This does not imply that demand

can never exceed the bound, but this bound reflects the maximum amount of demand that a

company wants to satisfy from safety stock. This interpretation is consistent with most

practical applications that safety stocks are used to cope with regular demand variations that do

not exceed a maximum reasonable bound. Larger demand variations are handled with

extraordinary measures, such as spot-market purchases, rescheduling or expediting orders,

other than using safety stocks. Thus, under these assumptions, each node sets its base stock so

as to meet all the orders from its downstream customers within the guaranteed service time.

That is, for node j, there is an associated safety stock factor jλ , which is given and corresponds

to the maximum amount of demand that company wants to satisfy from safety stock. This

yields the base stock level at node j:

j j j j j jS NLT NLTμ λ σ= +

This formula is similar but slightly different from the single stage inventory model in terms of

the expression for the lead time. Note that the review period has been taken into account as part

of the order processing time and considered in the net lead time.

With the guaranteed service approach, the total inventory cost consists of safety stock cost

and pipeline inventory cost. The safety stock of node j ( jSS ) is given by the following formula

as discussed above,

j j j jSS NLTλ σ=

The expected pipeline inventory is the sum of expected on hand and on-order inventories.

Based on Little’s law,33 the expected pipeline inventory jPI of node j equals to the mean

demand over the order processing time, and is given by,

j j jPI t μ=

which is not affected by the coverage and guaranteed service time decisions.

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3. Problem Statement We are given a potential supply chain (Figure 5) consisting of a set of plants (or suppliers)

i I∈ , a number of candidate sites for distribution centers j J∈ , and a set of customer

demand zones k K∈ whose inventory costs should be taken into account. The customer

demand zone can represent a local distributor, a regional warehouse, a dealer, a retailer, or a

wholesaler, which is usually a necessary component of the supply chain for specialty chemicals

or advanced materials.19, 34 Alternatively, one might view the customer demand as the

aggregation of a group of customers operated with vendor managed inventory (vendor takes

care of customers’ inventory), which is a common business model in the industrial gases

industry and some chemical companies.35, 36

[Figure 5]

In the given potential supply chain, the locations of the plants, potential distribution

centers and customer demand zones are known and the distances between them are given. The

investment costs for installing DCs are expressed by a cost function with fixed charges. Each

retailer i has an uncorrelated normally distributed demand with mean iμ and variance 2iσ

in each unit of time. Single sourcing restriction, which is common in the specialty chemicals

supply chain,37 is employed for the distribution from plants to DCs and from DCs to customer

demand zones. That is, each DC is only served by one plant, and each customer demand zone is

only assigned to one DC to satisfy the demand. Linear transportation costs are incurred for

shipments from plant i to distribution center j with unit cost 1ijc , and from distribution

center j to customer demand zone k with unit cost 2 jkc . The corresponding deterministic

order processing times of DCs and customer demand zone that includes the material handling

time, transportation time and inventory review period, are given by 1ijt and 2 jkt . The service

time of each plant, and the maximum service time of each customer demand zones are known.

We are also given the safety stock factor for DCs and customer demand zones, 1 jλ and 2kλ ,

which correspond to the standard normal deviate of the maximum amount of demand that the

node will satisfy from its safety stock. A common review period is used for the control of

inventory in each node. Inventory costs are incurred at distribution centers and customers, and

consist of pipeline inventory and safety stock, of which the unit costs are given.

The objective is to determine how many distribution centers (DCs) to install, where to

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locate them, which plants to serve each DC and which DCs to serve each customer demand

zone, how long should each DC quote its service time, and what level of safety stock to

maintain at each DC and customer demand zone so as to minimize the total installation,

transportation, and inventory costs.

4. Model Formulation The joint multiechelon supply chain design and inventory management model is a

mixed-integer nonlinear program (MINLP) that deals with the supply chain network design for

a given product, and considers its multi-echelon inventory management. The definition of sets,

parameters, and variables of the model is given below:

Sets/Indices

I Set of plants (suppliers) indexed by i

J Set of candidate DC locations indexed by j

K Set of customer demand zones (wholesaler, regional distributor, dealer, retailer, or customers with vendor managed inventory) indexed by k

Parameters

1ijc Unit transportation cost from plant i to DC j

2 jkc Unit transportation cost from DC j to customer demand zone k

jf Fixed cost of installing a DC at candidate location j (annually)

jg Variable cost coefficient of installing candidate DC j (annually)

1jh Unit inventory holding cost at DC j (annually)

2kh Unit inventory holding cost at customer demand zone k (annually)

kR Maximum guaranteed service time to customers at customer demand zone k

iSI Guaranteed service time of plant i

1ijt Order processing time of DC j if it is served by plant i, including material handling time of DC j, transportation time from plant i to DC j, and inventory review period

2 jkt Order processing time of customer demand zone k if it is served by DC j, including material handling time of DC j, transportation time from DC j to customer demand zone k, and inventory review period

kμ Mean demand at customer demand zone k (daily)

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2kσ Variance of demand at customer demand zone k (daily)

χ Days per year (to convert daily demand and variance values to annual costs)

1ijθ Unit cost of pipeline inventory from plant i to DC j (annual)

2 jkθ Unit cost of pipeline inventory from DC j to customer demand zone k (annual)

1jλ Safety stock factor of DC j

2kλ Safety stock factor of customer demand zone k

Binary Variables (0-1)

ijX 1 if DC j is served by plant i , and 0 otherwise

jY 1 if we install a DC in candidate site j , and 0 otherwise

jkZ 1 if customer demand zone k is served by DC j , and 0 otherwise

Continuous Variables (0 to+∞ )

kL Net lead time of customer demand zone k

jN Net lead time of DC j

kR Guaranteed service time of customer demand zone k

jS Guaranteed service time of DC j to its successive customer demand zones

4.1. Objective Function

The objective of this model is to minimize the total supply chain design cost including the

following items:

• Installation costs of DCs

• transportation costs from plants to DCs and from DCs to customer demand zones

• pipeline inventory costs in DCs and customer demand zones

• safety stock costs in DCs and customer demand zones

The cost of installing a DC in candidate location j is expressed by a fixed-charge cost

model that captures the economies of scale in the investment. The annual expected demand of

DC j is ( jk kk KZχ μ

∈∑ ), which equals to the annual mean demand of all the customer demand

zones served by DC j. Hence, the cost of installing DC j consists of fixed cost jf and variable

cost ( j jk kk Kg Zχ μ

∈∑ ), which is the product of variable cost coefficient and the expected

demand of this DC in one year. Thus, the total installation costs of all the DCs is given by,

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j j j jk kj J j J k K

f Y g Zχ μ∈ ∈ ∈

⎛ ⎞+ ⎜ ⎟⎝ ⎠

∑ ∑ ∑ (1)

The product of the annual mean demand of DC j ( jk kk KZχ μ

∈∑ ) and the unit

transportation cost ( 1ij iji Ic X

∈∑ ) between DC j and the plant that serves it yields the annual

plant to DC transportation cost.

1ij ij jk ki I j J k K

c X Zχ μ∈ ∈ ∈

⎛ ⎞⎜ ⎟⎝ ⎠

∑∑ ∑ (2)

Similarly, the product of yearly expected mean demand of customer demand zone k ( kχμ )

and the unit transportation cost ( 2 jk jkj Jc Z

∈∑ ) between customer demand zone k and the DC

that serves it, leads to the annual DC to customer demand zone transportation cost.

2 jk jk kj J k K

c Zχ μ∈ ∈∑∑ (3)

Based on Little’s law,33 the pipeline inventory jPI of DC j equals to the product of its

daily mean demand ( jk kk KZ μ

∈∑ ) and its order processing time ( 1ij iji It X

∈∑ ), which is in

terms of days. Thus, the annual total pipeline inventory cost of all the DCs is given by,

1 1j ij ij jk ki I j J k K

t X Zθ μ∈ ∈ ∈

⎛ ⎞⎜ ⎟⎝ ⎠

∑∑ ∑ (4)

where 1jθ is the annual unit pipeline inventory holding cost of DC j.

Similarly, the total annual pipeline inventory cost of all the customer demand zones is

given by,

2 2k jk jk kj J k K

t Zθ μ∈ ∈∑∑ (5)

where 2kθ is the annual unit pipeline inventory holding cost of customer demand zone k, and

2 jk jk kk Kt Z μ

∈∑ is the pipeline inventory of customer demand zone k.

The demand at customer demand zone k follows a given normal distribution with mean

kμ and variance 2kσ . Due to the risk-pooling effect,31 the demand over the net lead time ( jN )

at DC j is also normally distributed with a mean of k

j kk JN μ

∈∑ and a variance of

2

kj kk J

N σ∈∑ , where kJ is the set of customer demand zones k assigned to DC j. Thus, the

safety stock required in the DC at candidate location j with a safety stock factor 1 jλ is

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21 j j k jkk K

N Zλ σ∈

⋅ ∑ . Considering the annual inventory holding cost at DC j is 1 jh , we have

the annual total safety stock cost at all the DCs equals to

21 1j j j k jkj J k K

h N Zλ σ∈ ∈

⋅∑ ∑ (6)

Similarly, the demand over the net lead time of customer demand zones k ( kL ) is normally

distributed with a mean of k kL μ and a variance of 2k kL σ . Thus, the annual safety stock cost at

all the customer demand zones is given by,

2 2k k k kk K

h Lλ σ∈

⋅∑ (7)

Therefore, the objective function of this model (the total supply chain design cost) is given

by

2

min :

1 2

1 1 2 2

1 1 2 2

j j j jk kj J j J k K

ij ij jk k jk jk ki I j J k K j J k K

j ij ij jk k k jk jk ki I j J k K j J k K

j j j k jk k k k kj J k K k

f Y g Z

c X Z c Z

t X Z t Z

h N Z h L

χ μ

χ μ χ μ

θ μ θ μ

λ σ λ σ

∈ ∈ ∈

∈ ∈ ∈ ∈ ∈

∈ ∈ ∈ ∈ ∈

∈ ∈ ∈

⎛ ⎞+ ⎜ ⎟⎝ ⎠

⎛ ⎞+ +⎜ ⎟⎝ ⎠

⎛ ⎞+ +⎜ ⎟

⎝ ⎠

+ ⋅ + ⋅

∑ ∑ ∑

∑∑ ∑ ∑∑

∑∑ ∑ ∑∑

∑ ∑K∑

(8)

where each term accounts for the DC installation cost, transportation costs of plants-DCs and

DCs-customer demand zones, pipeline inventory costs of DCs and customer demand zones,

and safety stock costs of DCs and customer demand zones.

4.2. Constraints

Three constraints are used to define the network structure. The first one is that if DC j is

installed, it should be served by only one plant. If it is not installed, it is not assigned to any

plant. This can be modeled by,

ij ji I

X Y∈

=∑ , j∀ (9)

The second constraint states that each customer demand zone k is served by only one DC,

1jkj J

Z∈

=∑ , k∀ (10)

The third constraint is that if a customer demand zone k is served by the DC in candidate

location j , the DC must exist,

jk jZ Y≤ , ,j k∀ (11)

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Two constraints are used to define the net lead time of DCs and customer demand zones.

The replenishment lead time of DC j should be equal to the guaranteed service time ( iSI ) of

plant i, which serves DC j, plus the order processing time ( ijt ). Since each DC is served by only

one plant, the replenishment lead time of DC j is calculated by ( )i ij iji ISI t X

∈+ ⋅∑ . Thus, the

net lead time of DC j should be greater than its replenishment lead time minus its guaranteed

service time to its successor customer demand zones, and is given by the linear inequality,

( )∈

≥ + ⋅ −∑j i ij ij ji I

N SI t X S , j∀ (12)

Similarly, the net lead time of a customer demand zone k is greater than its replenishment

lead time minus its maximum guaranteed service time, kR , which is given by the nonlinear

inequalitiy

( )∈

≥ + ⋅ −∑k j jk jk kj J

L S t Z R , k∀ (13)

Finally, all the decision variables for network structure are binary variables, and the

variables for guaranteed service time and net lead time are non-negative variables.

, , {0,1}ij j jkX Y Z ∈ , , ,i j k∀ (14)

0≥jS , 0≥jN , ∀j (15)

0≥kL , k∀ (16)

4.3. MINLP Model

Grouping the parameters, we can rearrange the objective function and formulate the problem as

the following mixed-integer nonlinear program (P0):

Min: 21 2j j ijk ij jk jk jk j j k jk k kj J i I j J k K j J k K j J k K k K

f Y A X Z B Z q N Z q Lσ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

+ + + +∑ ∑∑∑ ∑∑ ∑ ∑ ∑ (17)

s.t. ij ji I

X Y∈

=∑ , j∀ (9)

1jkj J

Z∈

=∑ , k∀ (10)

jk jZ Y≤ , ,j k∀ (11)

∈

≥ ⋅ −∑j ij ij ji I

N S X S , j∀ (12)

( )∈

≥ + ⋅ −∑k j jk jk kj J

L S t Z R , k∀ (13)

, , {0,1}ij j jkX Y Z ∈ , , ,i j k∀ (14)

-16-

0≥jS , 0≥jN , ∀j (15)

0≥kL , k∀ (16)

where

= +ij i ijS SI t

( )1 1 1χ θ μ= + ⋅ijk ij j ij kA c t

( )2 2 2χ χ θ μ= + + ⋅jk j jk k jk kB g c t

1 1 1j j jq hλ= ⋅

2 2 2k k k kq hλ σ= ⋅ ⋅

4.4. Reformulated MINLP Model

The MINLP model (P0) has nonconvex terms, including bilinear and square root terms, in

its objective function (17) and constraint (13). Because the bilinear terms are the products of

two binary variables (such as ij jkX Z⋅ ), or the product of a binary variable and a continuous

variable (such as j jkN Z⋅ and j jkS Z⋅ ), they can be linearized by introducing additional

variables.

To linearize the term ( ij jkX Z⋅ ) in the objective function (17), we first introduce a new

continuous non-negative variable ijkXZ . Then the product of ijX and jkZ can be replaced by

this term ijkXZ with the following constraints,38

≤ijk ijXZ X , , ,i j k∀ (19)

≤ijk jkXZ Z , , ,i j k∀ (20)

1≥ + −ijk ij jkXZ X Z , , ,i j k∀ (21)

0ijkXZ ≥ , , ,i j k∀ (22)

where constraints (19), (20) and (22) ensure that if ijX or jkZ is zero, ijkXZ should be zero,

and constraint (21) ensures that if ijX and jkZ are both equal to one, ijkXZ should be one.

The linearization of ( j jkS Z⋅ ) in constraint (13) requires two new continuous non-negative

variable jkSZ and 1jkSZ , and the following constraints,38

1+ =jk jk jSZ SZ S , ,j k∀ (23)

≤ ⋅ Ujk jk jSZ Z S , ,j k∀ (24)

-17-

1 (1 )≤ − ⋅ Ujk jk jSZ Z S , ,j k∀ (25)

0≥jkSZ , 1 0≥jkSZ , ,j k∀ (26.1)

where constraints (24), (25) and (26.1) ensure that if jkZ is zero, jkSZ should be zero; if jkZ

is one, 1jkSZ should be zero. Combining with constraint (23), we can have jkSZ equivalent

to the product of jS and jkZ .

Similarly, the product of jN and jkZ in the objective function (17) can be linearized as

follows,

1+ =jk jk jNZ NZ N , ,j k∀ (27)

≤ ⋅ Ujk jk jNZ Z N , ,j k∀ (28)

1 (1 )≤ − ⋅ Ujk jk jNZ Z N , ,j k∀ (29)

0≥jkNZ , 1 0≥jkNZ , ,j k∀ (26.2)

where jkNZ and 1 jkNZ are two new continuous variables, and jkNZ is equivalent to

( j jkN Z⋅ ).

The above linearizations introduce more constraints and variables, but significantly

reduces the number of nonlinear terms in the model (P0) and potentially reduce the

computational effort. To further reduce the nonlinear terms in the objective function (17), the

term ( 2j k jkk K

N Zσ∈

⋅ ⋅∑ ) is replaced by a new nonnegative continuous variable jNZV with

the following constraint, 2σ

∈

= ⋅∑j k jkk K

NZV NZ , j∀ (30)

0jNZV ≥ , j∀ (31)

Therefore, incorporating the above linearizations we have the following reformulated

MINLP model (P1).

Min: 1 2j j ijk ijk jk jk j j k kj J i I j J k K j J k K j J k K

f Y A XZ B Z q NZV q L∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

+ + + +∑ ∑∑∑ ∑∑ ∑ ∑ (17)

s.t. ij ji I

X Y∈

=∑ , j∀ (9)

1jkj J

Z∈

=∑ , k∀ (10)

jk jZ Y≤ , ,j k∀ (11)

-18-

∈

≥ ⋅ −∑j ij ij ji I

N S X S , j∀ (18)

∈ ∈

≥ + ⋅ −∑ ∑k jk jk jk kj J j J

L SZ t Z R , k∀ (32)

≤ijk ijXZ X , , ,i j k∀ (19)

≤ijk jkXZ Z , , ,i j k∀ (20)

1≥ + −ijk ij jkXZ X Z , , ,i j k∀ (21)

1+ =jk jk jSZ SZ S , ,j k∀ (23)

≤ ⋅ Ujk jk jSZ Z S , ,j k∀ (24)

1 (1 )≤ − ⋅ Ujk jk jSZ Z S , ,j k∀ (25)

1+ =jk jk jNZ NZ N , ,j k∀ (27)

≤ ⋅ Ujk jk jNZ Z N , ,j k∀ (28)

1 (1 )≤ − ⋅ Ujk jk jNZ Z N , ,j k∀ (29)

2σ∈

= ⋅∑j k jkk K

NZV NZ , j∀ (30)

, , {0,1}ij j jkX Y Z ∈ , , ,i j k∀ (14)

0≥jS , 0≥jN , ∀j (15)

0≥kL , k∀ (16)

0ijkXZ ≥ , , ,i j k∀ (22)

0jNZV ≥ , j∀ (31)

0≥jkSZ , 1 0≥jkSZ , 0≥jkNZ , 1 0≥jkNZ , ,j k∀ (26)

Compared with model (P0), model (P1) is computationally more tractable, because all the

constraints in (P1) are linear and the only nonlinear terms are univariate concave terms in the

objective function.

4.5. Variable Bounds and Initialization

The bounds of variables are quite important for nonlinear optimization problems. The

upper bounds of the key continuous variables are derived below (denoted with the superscript

U).

The maximum net lead time ( UjN ) of DC j is the maximum value of the sum of service

time of plant i and the order processing time from plant i to DC j. It means that when the

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guaranteed service time of DC j is zero and it is served by plant i that has the maximum

( i ijSI t+ ), the net lead time jN equals to the maximum value,

{ } { }max max∈ ∈

= + =Uj i ij iji I i I

N SI t S , j∀ (33.1)

The maximum value of guaranteed service time of DC j is equal to the maximum value of

its net lead time. It means that when the net lead time of DC j is zero and it is assigned to plant

i that has the maximum ( i ijSI t+ ), the net lead time jS equals to the maximum value,

{ }maxU Uj j iji I

S N S∈

= = , j∀ (33.2)

From constraints (13) and (16), it is easy to see that the maximum net lead time of

customer demand zone k is as follows,

{ }max 0,max∈

⎧ ⎫= + −⎨ ⎬⎩ ⎭

U Uk j jk k

j JL S t R , j∀ (33.3)

The upper bounds of the auxiliary variables are easy to derive,

=U Ujk jSZ S , 1 =U U

jk jSZ S , ,j k∀ (33.4)

=U Ujk jNZ N , 1 =U U

jk jNZ N , ,j k∀ (33.5)

2σ∈

= ⋅∑U Uj k j

k KNZV N , j∀ (33.6)

Besides variable bounds, the initial point is another key issue for numerical optimization of

MINLP problems. Since model (P1) has some special structure (linear constraints, univariate

concave terms in the objective function), we can use a similar approach as in You and

Grossmann30 to obtain a “good” starting point by solving a mixed-integer linear program

(MILP) problem for initialization purposes.

[Figure 6]

As introduced by Falk and Soland,39 for a univariate square root term x , where the

variable x has lower bound 0 and upper bound Ux , its secant / Ux x represents the convex

envelope and provides a valid lower bound of the square root term as shown in Figure 6. Since

model (P1) is a minimization problem and all the constraints are linear, replacing all the

univariate square root terms with their secants in the objective function (17) will lead to the

MILP problem (P2) with a linear objective function,

Min: 1 2j j k k

j j ijk ijk jk jk U Uj J i I j J k K j J k K j J k Kj k

q NZV q Lf Y A XZ B ZNZV L∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

⋅ ⋅+ + + +∑ ∑∑∑ ∑∑ ∑ ∑ (34)

-20-

s.t. all the constraints of (P1)

Since (P1) and (P2) have the same constraints, a feasible solution of (P2) is also a

feasible solution of (P1). Furthermore, the optimal objective function value of (P2) is a valid

lower bound of the optimal objective function value of (P1). Therefore, by solving the

initialization MILP problem (P2), we can obtain a “good” starting point of solving problem

(P1).

Note that based on this initialization procedure, a heuristic algorithm is to first solve

problem (P2), then using its optimal solution as an initial point to solve problem (P1) with an

MINLP solver that relies on convexity assumption (e.g. DICOPT, SBB, etc.). This algorithm

can obtain “good” solutions quickly, although global optimality cannot be guaranteed.

5. Illustrative Example To illustrate the application of this model, we consider two small illustrative examples, an

industrial liquid oxygen (LOX) supply chain example and an acetic acid supply chain example.

5.1. Example A: Industrial LOX Supply Chain

The first example is for an industrial LOX supply chain with two plants, three potential

DCs and six customers as given in Figure 7. In this supply chain, customer inventories are

managed by the vendor, i.e. vendor-management-inventory (VMI), which is a common

business model in the gas industry.34-36 Thus, in this example inventory costs from DCs and

customers should be taken into account into the total supply chain cost, and the joint

multiechelon supply chain design and inventory management model can be used to minimize

total network design, transportation and inventory costs.

[Figure 7] In this instance, the annual fixed costs to install the DCs ( jf ) are $10,000/year,

$8,000/year and $12,000/year, respectively. The variable cost coefficient of installing a DC

( jg ) at all the candidate locations is $0.01/(liter · year). The safety stock factors for DCs ( 1jλ )

and customers ( 2kλ ) are the same and equal to 1.96, which corresponds to 97.5% service level

if demand is normally distributed. We consider 365 days in a year ( χ ). The guaranteed service

times of the two plants ( iSI ) are 2 days and 3 days, respectively. Since the last echelon

represents the customers, the guaranteed service time of customers ( kR ) are set to 0. The data

-21-

for demand uncertainty, order processing times and transportation costs are given in Tables

1-5.

[Table 1] [Table 2] [Table 3] [Table 4] [Table 5]

We consider two instances of this example. In the first instance, we consider zero holding

cost for both pipeline inventory and safety stocks, i.e. the problem reduces to a supply chain

network design problem without considering inventory cost. In the second instance, the

pipeline inventory holding cost of LOX is $6/(liter · year) for all the DCs ( 1ijθ ) and customers

( 2 jkθ ), and the safety stock holding cost is $8/(liter · year) for all the DCs ( 1jh ) and customers

( 2kh ).

Both the initialization MILP model (P2) and the MINLP model (P1) include 27 binary

variables, 124 continuous variables and 256 constraints. Since the problem sizes are relatively

small, we solve model (P1) directly to obtain the global optimum with 0% optimality margin

by using the BARON solver40 with GAMS,41 and the initialization MILP model (P2) is solved

with GAMS/CPLEX. The resulting optimal supply chain networks with a minimum annual

cost of $108,329/year and the flow rate of each transportation link are given in Figure 8. We

can see that for the instance without considering inventory costs, all the three DCs are installed,

and each of them serves two customers. The major trade-off is between DC installation costs

and the transportation costs. For the instance that takes into account the inventory costs, only

two DCs are installed and each serves three customers. The major trade-off is between DC

installation costs, transportation costs and inventory costs. The minimum annual cost in this

instance is $152,107/year with the components given in Figure 9. A comparison between these

two instances suggests the importance of integrating inventory costs in the supply chain

network design. Although the inventory costs might only make up a small portion of the total

costs, the optimal network structure could be quite different with and without considering

inventory in the supply chain design.

[Figure 8] [Figure 9]

-22-

5.2. Example B: Acetic Acid Supply Chain

The second example is for an acetic acid supply chain with three plants, three potential

DCs and four customer demand zones, each of which has a wholesaler (Figure 10). Based on

the conclusion of the previous example, we need to take into account the inventory costs from

DCs and wholesalers when designing this supply chain, so as to obtain a more accurate

“optimal” supply chain.

[Figure 10] In this instance, the annual fixed DC installation cost ( jf ) is $50,000/year for all the DCs.

The variable cost coefficient of installing a DC ( jg ) at all the candidate locations is $0.5/(ton ·

year). The safety stock factors for DCs ( 1jλ ) and wholesalers ( 2kλ ) are the same and equal to

1.96. We consider 365 days in a year ( χ ). The guaranteed service times of the three vendors

( iSI ) are 3 days, 3 days and 4 days, respectively. The pipeline inventory holding cost is $1/(ton

· day) for all the DCs ( 1 /ijθ χ ) and wholesalers ( 2 /jkθ χ ), and the safety stock holding cost is

$1.5/(ton · day) for all the DCs ( 1 /jh χ ) and wholesalers ( 2 /kh χ ). Since the guaranteed

service time of wholesalers to end customers ( kR ) are set to be 8 days. The data for demand

uncertainty, order processing times and transportation costs are given in Tables 6-10.

[Table 6] [Table 7] [Table 8] [Table 9] [Table 10]

The computational study is carried out on an IBM T40 laptop with Intel 1.50GHz CPU and

512 MB RAM. The original model (P0) has 12 discrete variables, 22 continuous variables and

27 constraints. Both the initialization MILP model (P2) and the MINLP model (P1) include 24

binary variables, 185 continuous variables and 210 constraints. We solve the original model

(P0) by using GAMS/BARON with 0% optimality margin, and it takes a total of 1376.9 CPU

seconds. We then solve the reformulated model (P1) with the aforementioned initialization

process, the CPU time reduces to only 7.6 seconds and the optimal solutions are the same as we

obtained by solving model (P0). The possible reason is that initialization process helps

BARON to find a “good” feasible solution during the preprocessing step. This feasible solution

provides a tighter upper bound that reduces the searching space and speeds up the computation.

-23-

The resulting optimal supply chain networks with a minimum annual cost of $

1,986,148/year and the flow rate of each transportation link are given in Figure 11. We can see

that two DCs are selected to install and one of them serves three wholesalers while the other

one only serves one wholesaler. The major trade-off is between DC installation costs,

transportation costs and inventory costs. A detailed breakdown of the total cost is given in

Figure 12.

[Figure 11] [Figure 12]

6. Solution Algorithm Although small scale problems can be solved to global optimality effectively by using a

global optimizer and the aforementioned initialization method, medium and large-scale joint

multi-echelon supply chain design and inventory management problems are often

computationally intractable with direct solution approaches due to the combinatorial nature

and nonlinear nonconvex terms. In this section, we present an effective solution algorithm

based on Lagrangean relaxation and piecewise linear approximation to obtain solutions within

1% of global optimality gap with reasonable computational expense.

6.1. Piecewise Linear Approximation

Instead of using the secants in the objective function of model (P1), a tighter lower convex

envelope of the univariate square root terms is the piecewise linear function, which employs a

few more continuous and discrete variables. Although this may require longer computational

times to solve the initialization MILP problem, with the significant progress of MILP solvers,

piecewise linear approximations have recently been increasingly used for approximating

different types of nonconvex nonlinear functions,42, 43 especially for univariate concave

functions.20, 44-46

There are several different approaches to model the piecewise linear function for a concave

term.20, 43, 47 In this work, we use the well-known “multiple-choice” formulation20 to

approximate the square root term x . Let { }1,2,3, ,P p= ⋅⋅⋅ denote the set of intervals in the

piecewise linear function ( )xϕ , and 1M , 2M ,… pM , 1pM + be the upper and lower bounds

of x for each interval p. The “multiple choice” formulation of ( )xϕ is then given by,

( ) min ( )p p p pp

x F v C uϕ = +∑

s.t. 1pp

v =∑

-24-

pp

u x=∑

1p p p p pM v u M v+≤ ≤ , p P∈

{ }0,1pv ∈ , p P∈

where 1

1

p pp

p p

M MC

M M+

+

−=

− and p p p pF M C M= − , p P∈ .

[Figure 13]

As can be seen in Figure 13, the more intervals are used, the better is the approximation of

the nonlinear function, but more additional variables and constraints are required. Using this

“multiple choice” formulation to approximate the univariate concave terms jNZV and

kL in the objective function of problem (P1), yields the following piecewise linear MILP

problem (P3), which provides a tighter lower bounding problem of (P1).

(P3):

( ) ( )1 1 1 1 2 2 2 2

1 2

, , , , , , , ,

Min:

1 1 1 1 1 2 2 2 2 2

j j ijk ijk jk jkj J i I j J k K j J k K

j j p j p j p j p k k p k p k p k pj J p k K p

f Y A XZ B Z

q F v C u q F v C u∈ ∈ ∈ ∈ ∈ ∈

∈ ∈

+ +

+ + + +

∑ ∑∑∑ ∑∑

∑ ∑ ∑ ∑(35)

s.t. 1

1

,1 1j pp

v =∑ , j∀ (36.1)

1

1

,1 j p jp

u NZV=∑ , j∀ (36.2)

1 1 1 1 1, , , , 1 ,1 1 1 1 1j p j p j p j p j pM v u M v+≤ ≤ , 1,j p∀ (36.3)

{ }1,1 0,1j pv ∈ ,

1,1 0j pu ≥ , 1,j p∀ (36.4)

2

2

,2 1k pp

v =∑ , k∀ (37.1)

2

2

,2k p kp

u L=∑ , k∀ (37.2)

2 2 2 2 2, , , , 1 ,2 2 2 2 2k p k p k p k p k pM v u M v+≤ ≤ , 2,k p∀ (37.3)

{ }2,2 0,1k pv ∈ ,

2,2 0k pu ≥ , 2,k p∀ (37.4)

All the constraints of (P1).

Note that any feasible solution obtained from the problem (P3) is also a feasible solution of

(P1), and for each feasible solution the objective function value of (P3) is always less than or

-25-

equal to the objective function value of (P1).

In summary, the solution obtained by solving (P3) can provide a “good” initial point of

solving problem (P1). On the other hand, by substituting the solution of (P3) into the objective

function of problem (P1), i.e. function evaluation, or using that solution as an initial point to

solve (P1) with a nonlinear programming (NLP) solver with fixed integer variables that are

solution of (P3) or MINLP solver. In addition, we can obtain a valid upper bound of the

“global” optimal objective function value of (P1).

In order to obtain an initial point “close” enough to the global solution, we can in principle

use piecewise linear functions with sufficient large number of intervals to approximate the

univariate concave terms in (P1). However, it is a nontrivial task to solve large-scale instances

of (P3). To solve the problem effectively, we first exploit some properties of problem (P1).

6.2. Alternative Formulation

Let us first consider an alternative model formulation (AP) of problem (P1), in which

(23)–(25) are excluded and represent the inequality (13) by a disaggregated disjunction (39),

Min: 1 2j j ijk ijk jk jk j j k jkj J i I j J k K j J k K j J j J k K

f Y A XZ B Z q NZV q L∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

+ + + +∑ ∑∑∑ ∑∑ ∑ ∑∑ (38)

s.t. ij ji I

X Y∈

=∑ , j∀ (9)

1jkj J

Z∈

=∑ , k∀ (10)

jk jZ Y≤ , ,j k∀ (11)

∈

≥ ⋅ −∑j ij ij ji I

N S X S , j∀ (18)

0¬⎡ ⎤ ⎡ ⎤

∨⎢ ⎥ ⎢ ⎥≥ + − ≤⎣ ⎦ ⎣ ⎦

jk jk

jk j jk k jk

Z ZL S t R L

, ,j k∀ (39)

≤ijk ijXZ X , , ,i j k∀ (19)

≤ijk jkXZ Z , , ,i j k∀ (20)

1≥ + −ijk ij jkXZ X Z , , ,i j k∀ (21)

1+ =jk jk jNZ NZ N , ,j k∀ (27)

≤ ⋅ Ujk jk jNZ Z N , ,j k∀ (28)

1 (1 )≤ − ⋅ Ujk jk jNZ Z N , ,j k∀ (29)

-26-

2σ∈

= ⋅∑j k jkk K

NZV NZ , j∀ (30)

, , {0,1}ij j jkX Y Z ∈ , , ,i j k∀ (14)

0≥jS , 0≥jN , ∀j (15)

0jkL ≥ , ,j k∀ (40)

0ijkXZ ≥ , , ,i j k∀ (22)

0jNZV ≥ , j∀ (31)

Note that the set of disjunction in (39) is written in terms of the disaggregated variable

jkL , which is then substituted in the last term of the objective function (38). Applying the

convex hull reformulation48, 49 to the above disjunctive constraint (39) in (AP) leads to:

≤ ⋅ Ujk jk jkL Z L , ,j k∀ (41.1)

1≥ + ⋅ − ⋅jk jk jk jk k jkL S t Z R Z , ,j k∀ (41.2)

1 2= +j jk jkS S S , ,j k∀ (41.3)

1 ≤ ⋅ Ujk jk jS Z S , ,j k∀ (41.4)

2 (1 )≤ − ⋅ Ujk jk jS Z S , ,j k∀ (41.5)

1 0≥jkS , 2 0≥jkS , ,j k∀ (41.6)

where 1 jkS and 2 jkS are two new auxiliary variables.

The value of the new continuous variable jkL in (AP) is defined through two new

constraints (39) and (40). It means that if customer demand zone k is assigned to DC j, i.e.,

1jkZ = , then the new variable jkL represents the net lead time of customer demand zone k,

otherwise 0jkL = . Thus, an important proposition is that any feasible solution of problem (AP)

is also a feasible solution of problem (P1), and vice versa.

Proposition 1. If ( * * * * * *, , , , ,ij j jk j j kX Y Z S N L ) is a feasible solution of problem (P1) with objective

function value *W , then ( * * * * * *, , , , ,ij j jk j j jkX Y Z S N L ), where * 0jkL = if * 0jkZ = and * *jk kL L= if

* 1jkZ = , is a feasible solution of problem (AP) and the associated objective function value of

(AP) is *W . If ( * * * * * *, , , , ,ij j jk j j jkX Y Z S N L ) is a feasible solution of problem (AP) with objective

-27-

function value *W , then ( * * * * * *, , , , ,ij j jk j j kX Y Z S N L ), where * *k jkj J

L L∈

=∑ , is a feasible solution

of problem (P1) and the associated objective function value of (P1) is *W

The proof of this proposition is given in the appendix. Proposition 1 shows the

equivalence between (P1) and (AP). We can solve one of these two problems to obtain a

feasible or optimal solution and then obtain the solution of another problem with the same

objective function value after some algebraic derivations. Compared to model (P1), model

(AP) has more square root terms in the objective function due to the introduction of variable

jkL , although the linearization constraints (23), (24), (25) for the product of jS and jkZ are

not included in (AP).

It might not be a wise idea to solve (AP) with direct solution approach to obtain an

optimal solution of (P1) since there are more nonlinear terms in (AP). However, (AP) can be

decomposed into DCs with which a Lagrangean relaxation algorithm as will be shown in the

next section can be used to speed up the computation. Further, because of the equivalence of

(P1) and (AP), we can solve (P1) instead of (AP) in the full or reduced variable space during

the solution procedure of the Lagrangean algorithm. In the next section, we describe how to

incorporate model (AP) and the piecewise linear approximation into a Lagrangean relaxation

algorithm.

6.3. Lagrangean Relaxation Algorithm

In order to obtain near global optimal solutions to problems (P1) and (AP) with modest

computational effort, we propose a decomposition algorithm based on Lagrangean relaxation.

6.3.1. The Decomposition Procedure

In this solution procedure, we use a decomposition scheme by dualizing the assignment

constraints (10) in (AP) using the Lagrangean multipliers kλ . As a result, we obtain the

following relaxed problem (APL),

( )= Min: 1 2

λ

λ∈ ∈ ∈ ∈ ∈

∈

⎛ ⎞+ + − + +⎜ ⎟⎝ ⎠

+

∑ ∑∑ ∑ ∑

∑

j j ijk ijk jk k jk j j k jkj J i I k K k K k K

kk K

W f Y A XZ B Z q NZV q L

(42)

s.t. ij ji I

X Y∈

=∑ , j∀ (9)

-28-

jk jZ Y≤ , ,j k∀ (11)

∈

≥ ⋅ −∑j ij ij ji I

N S X S , j∀ (18)

≤ijk ijXZ X , , ,i j k∀ (19)

≤ijk jkXZ Z , , ,i j k∀ (20)

1≥ + −ijk ij jkXZ X Z , , ,i j k∀ (21)

1+ =jk jk jNZ NZ N , ,j k∀ (27)

≤ ⋅ Ujk jk jNZ Z N , ,j k∀ (28)

1 (1 )≤ − ⋅ Ujk jk jNZ Z N , ,j k∀ (29)

2σ∈

= ⋅∑j k jkk K

NZV NZ , j∀ (30)

≤ ⋅ Ujk jk jkL Z L , ,j k∀ (41.1)

1≥ + ⋅ − ⋅jk jk jk jk k jkL S t Z R Z , ,j k∀ (41.2)

1 2= +j jk jkS S S , ,j k∀ (41.3)

1 ≤ ⋅ Ujk jk jS Z S , ,j k∀ (41.4)

2 (1 )≤ − ⋅ Ujk jk jS Z S , ,j k∀ (41.5)

1 0≥jkS , 2 0≥jkS , ,j k∀ (41.6)

, , {0,1}ij j jkX Y Z ∈ , , ,i j k∀ (14)

0≥jS , 0≥jN , ∀j (15)

0jkL ≥ , ,j k∀ (16)

0ijkXZ ≥ , , ,i j k∀ (22)

0jNZV ≥ , j∀ (31)

where W is the objective function value. Note that ( P(λ) ) can be decomposed into J

subproblems, one for each candidate DC site j J∈ , where each one is denoted by (APLj) and

is shown for a specific subproblem for candidate DC site j as follows:

( ) = Min: 1 2j j j ijk ijk jk k jk j j k jki I k K k K k K

W f Y A XZ B Z q NZV q Lλ∈ ∈ ∈ ∈

+ + − + +∑∑ ∑ ∑ (43)

s.t. ij ji I

X Y∈

=∑ ,

-29-

jk jZ Y≤ , k∀

∈

≥ ⋅ −∑j ij ij ji I

N S X S ,

≤ijk ijXZ X , ,i k∀

≤ijk jkXZ Z , ,i k∀

1≥ + −ijk ij jkXZ X Z , ,i k∀

1+ =jk jk jNZ NZ N , k∀

≤ ⋅ Ujk jk jNZ Z N , k∀

1 (1 )≤ − ⋅ Ujk jk jNZ Z N , k∀

2σ∈

= ⋅∑j k jkk K

NZV NZ ,

≤ ⋅ Ujk jk jkL Z L , k∀

1≥ + ⋅ − ⋅jk jk jk jk k jkL S t Z R Z , k∀

1 2= +j jk jkS S S , k∀

1 ≤ ⋅ Ujk jk jS Z S , k∀

2 (1 )≤ − ⋅ Ujk jk jS Z S , k∀

1 0≥jkS , 2 0≥jkS , k∀

, , {0,1}ij j jkX Y Z ∈ , ,i k∀

0≥jS , 0≥jN ,

0jkL ≥ , k∀

0ijkXZ ≥ , ,i k∀

0jNZV ≥ ,

Hence, (APL) can be decomposed into J subproblems (APLj), and one for each

candidate DC site j J∈ . Let jW denote the globally optimal objective function value of

problem (APLj). As a result of the decomposition procedure, the global minimum of (APL),

which corresponds to a lower bound of the global optimum of problem (AP), can be calculated

by:

-30-

j kj J k K

W W λ∈ ∈

= +∑ ∑ . (44)

For each fixed value of the Lagrangean multipliers kλ , we solve problem (APLj) for each

candidate DC location j . Then, based on (44), the optimal objective function value of

problem (APL) can be calculated for each fixed value of kλ . Using a standard subgradient

method50, 51 to update the Lagrangean multiplier kλ , the algorithm iterates until a preset

optimality tolerance is reached.

6.3.2. Lagrangean Relaxation Subproblems

In each iteration with fixed values of the Lagrangean multipliers kλ , the binary variables for

installing a DC ( jY ) are optimized separately in each subproblem (APLj) in the aforementioned

decomposition procedure. For each subproblem (APLj), we can observe that the objective

function value of (APLj) is 0 if and only if 0jY = (i.e. we do not install DC j ). In other words,

there is a feasible solution that leads to the objective of subproblem (APLj) equal to 0.

Therefore, the global minimum of subproblem (APLj) should be less than or equal to zero.

Given this observation, it is possible that for some value of kλ (such as 0kλ = , k K∈ ) the

optimal objective function values for all the subproblem (APLj) are 0 (i.e., 0jY = , ∈j J , we

do not select any DC). However, the original assignment constraint (8) implies that at least one

DC should be selected to meet the demands, i.e.

1jj J

Y∈

≥∑ . (45)

Once constraint (10) is relaxed, constraint (45) becomes “not redundant” and should be taken

into account in the algorithm.30, 52 To satisfy the constraint (45) in the Lagrangean relaxation

procedure, we make the following modifications to the aforementioned step of solving problem

(APLj) for each candidate DC location j .

First, consider the problem (APLRj), which is actually a special case of (APLj) when

1jY = . The formulation for a specific j is given as:

( ) = Min: 1 2j j ijk ijk jk k jk j j k jki I k K k K k K

W f A XZ B Z q NZV q Lλ∈ ∈ ∈ ∈

+ + − + +∑∑ ∑ ∑ (46)

s.t. the same constraints as (APLj) with 1jY = .

where jW is denoted as the optimal objective function value of the problem (APLRj).

With a similar scheme as introduced in our previous work,30 we use the following solution

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procedure to deal with the implied constraint (45): for each fixed value of kλ , we solve (APLRj)

for every candidate DC location j . Then select the DCs in candidate location j (i.e. let

1jY = ), for which 0jW ≤ . For all the remaining DCs for which 0jW > , we do not select

them and set 0jY = . If all the 0jW > , j J∀ ∈ , we select only one DC with the minimum jW ,

i.e. * 1jY = for the *j such that * min{ }j jj J

W W∈

= . By doing this at each iteration of the

Lagrangean relaxation (for each value of the multiplier iλ ), we ensure that the optimal solution

always satisfies 1jj J

Y∈

≥∑ . Thus, the globally optimal objective function of (APLj) can be

recalculated as:

, 1j

j kj J Y k K

W W λ∈ = ∈

= +∑ ∑ . (57)

Since (APLRj) includes 1k + univariate concave terms in the objective function solving

it to global optimality might be computationally expensive if |k| is large. To improve the

computational efficiency, piecewise linear approximations can be used. Similarly as in (P3),

we use the “multiple choice” formulation to approximate the concave terms in (APLRj), and

obtain the following MILP model (APLPj),

( )

( ) ( )1 1 1 1

1

3 3 3 3

, , , ,

, , , , , , , ,3

= Min: 1 1 1 1 1

2 3 3 3 3

j j ijk ijk j j p j p j p j pi I k K p

jk k jk k j k p j k p j k p j k pk K k K p

W f A XZ q F v C u

B Z q F v C uλ

∈ ∈

∈ ∈

+ + +

+ − + +

∑∑ ∑

∑ ∑ ∑ (48)

s.t. 1

1

,1 1j pp

v =∑ , (36.1)

11

,1 j p jp

u NZV=∑ , (36.2)

1 1 1 1 1, , , , 1 ,1 1 1 1 1j p j p j p j p j pM v u M v+≤ ≤ , 1p∀ (36.3)

{ }1,1 0,1j pv ∈ ,

1,1 0j pu ≥ , 1p∀ (36.4)

33

, ,3 1j k pp

v =∑ , k∀ (49.1)

33

, ,3 j k p jkp

u L=∑ , k∀ (49.2)

3 3 3 3 3, , , , , , , , 1 , ,3 3 3 3 3j k p j k p j k p j k p j k pM v u M v+≤ ≤ , 3,k p∀ (49.3)

{ }3, ,3 0,1j k pv ∈ ,

3, ,3 0j k pu ≥ , 3,k p∀ (49.4)

All the constraints of (APLRj).

-32-

where jW is the optimal objective function value of problem (APLPj).

As discussed in Section 6.1., jW provides a valid “global” lower bound of jW . Thus, the

globally optimal objective function value of problem (AP) has a globally lower bound W ,

which can be calculated by:

, 1j

j kj J Y k K

W W λ∈ = ∈

= +∑ ∑ . (50)

Based on this result, instead of solving each (APLRj) to globally optimality, we can use the

optimal solution of (APLPj) as an initial point to solve (APLRj) with an MINLP solver that

relies on convexity assumptions, such as DICOPT, SBB, etc., so as to improve the

computational efficiency. In this way, the optimal objective function of (APLj), calculated by

, 1j

j kj J Y k K

W W λ∈ = ∈

= +∑ ∑ , is no longer a valid global lower bound of the optimal objective

function value of problem (AP). However, the lower bound W , obtained from piecewise

linear lower bounding problem (APLPj), still provides a “globally” lower bound of the optimal

objective function value of problem (AP).

6.3.3. Upper Bounding

A feasible solution of problem (AP) naturally provides a valid upper bound of its global

optimal objective function value. To obtain a feasible solution, a common approach is to fix the

values of the binary variables ( ijX , jY and jkZ ) and solve (AP) in the reduced variable space

with an NLP solver. However, solving a large-scale NLP problem could be computationally

expensive.

Based on Proposition 1, to obtain a valid upper bound, we can solve problem (P1) instead

of (AP) with fixed values of binary variables ( ijX , jY and jkZ ) to reduce the computational

effort, because (P1) has fewer nonlinear terms than (AP). To avoid solving the large scale NLP

problem, we first solve the piecewise linear lower bounding MILP problem (P3) with fixed

binary variables, and then substitute the optimal solution into the objective function of problem

(P1) to calculate the associated objective function value. As discussed in Section 6.1., the

optimal solution of (P3) is a feasible solution of (P1). Thus, the objective function value of (P1)

obtained by function evaluation yields an upper bound of the global minimum of problem (AP).

By using this approach, we avoid using an NLP solver and improve the computational

efficiency and robustness without sacrificing the solution quality.

-33-

6.3.4. The Solution Algorithm

To summarize, the solution algorithm is as follows:

Step 1: (Initialization)

Use an arbitrary guess as the initial vector of Lagrange multipliers 1λ , or else the dual of

constraint (10) of a local optimum of the NLP relaxation of model (P1). Let the incumbent

upper bound be UB = +∞ , incumbent lower bound be LB = −∞ , global lower bound be

GLB = −∞ , and iteration number be 1t = . Set the step length parameter 2θ = .

Step 2:

With fixed Lagrange multipliers tλ , first solve the piecewise linear lower bounding

problem of the modified Lagrangean relaxation subproblem (APLPj) with an MILP solver.

Then fix the value of binary variables ijX and jkZ , use the optimal solution as the initial point

and solve the modified Lagrangean relaxation subproblem (APLRj) with an NLP solver (not

necessary global optimizer), or calculate the objective function value of (APLRj) directly using

the optimal solution of (APLPj).

Denote the optimal objective function value of (APLPj) as ( )t

jWλ

, the optimal objective

function value of (APLRj) as ( )t

jWλ

, and the optimal solutions of (APLRj) as (( ) ( )

,t tij jkX Zλ λ

).

If all ( )

0t

jWλ

> , j J∀ ∈ , let ( )* 1t

jY λ = , ( )( )

**t

t ijijX Xλλ = , for j* with { }( ) ( )

* mint t

j jj J

W Wλ λ

∈= .

Else, let ( ) 1tjY λ = ,

( )( ) tt ijijX X

λλ = for all j with ( )

0t

jWλ

≤ , and ( ) 0tjY λ = , ( ) 0t

ijX λ = for all j

such that ( )

0t

jWλ

> .

Calculate ( )

( )

, 1

t

tj

j kk Kj J Y

W Wλ

λ λ∈∈ =

= +∑ ∑ , ( )

( )

, 1

t

tj

j kk Kj J Y

W Wλ

λλ

∈∈ =

= +∑ ∑ .

If ( )t

W LBλ > , update lower bound by setting ( )t

LB W λ= . If more than 2 iterations of the

subgradient procedure50 are performed without an increment of LB , then halve the step length

parameter by setting 2θθ = .

If ( )tW GLBλ > , update lower bound by setting ( )tGLB W λ= .

-34-

Note that the strictly global lower bound is given by GLB instead of LB , if (APLRj) is

not solved with a global optimizer.

Step 3:

Fixing the design variable values as ( )tij ijX X λ= , ( )t

j jY Y λ= and using ( )tjkZλ

as the initial

values of the binary variables jkZ , solve the piecewise linear lower bounding problem (P3) in

the reduced space with fixed ijX , jY and tλ using an MILP solver. Denote the optimal

solution as ( ( )tjkZ λ , ( )t

jS λ , ( )tjN λ , ( )t

kL λ ). Substitute the optimal solution

( ( )tijX λ , ( )t

jY λ , ( )tjkZ λ , ( )t

jS λ , ( )tjN λ , ( )t

kL λ ) into problem (P1) and calculate its objective function value,

which is denoted as ( )t

Wλ

.

If ( )t

W UBλ

< , update the upper bound by setting ( )t

UB Wλ

= .

Step 4:

Calculate the subgradient ( kG ) using ( )

1 ttjkk

j J

G Zλ

∈

= −∑ , k∀ .

Compute the step size T ,50, 51 2

( )( )

tt

kk K

UB LBTG

θ

∈

⋅ −=

∑.

Update the multipliers, 1t t t tT Gλ λ+ = + ⋅ .

Step 5:

If UB LBgap tolUB−

= < (e.g. 10-3), or 21λ λ+ − <t t tol (e.g. 10-2) or the maximum

number of iterations has been reached, set UB as the optimal objective function value, and set

( ( )tijX λ , ( )t

jY λ , ( )tjkZ λ , ( )t

jS λ , ( )tjN λ , ( )t

kL λ ) as the optimal solution.

Else, increment t as 1t + , go to Step 2.

We should note that the entire procedure requires at least an MILP solver. An NLP solver

can be used to solve problem (APLRj) in the reduced variable space in Step 2, but the NLP

solver is not required. The reason is that the solution of the nonlinear optimization problem

(APLRj), which reduces to an NLP from MINLP after ijX and jkZ are fixed, can be

-35-

substituted by simple function evaluation as stated in Step 2, although the solution quality may

be sacrificed. Also, the above algorithm is guaranteed to provide rigorous global lower bounds

by GLB as discussed in Step 2. Thus, the gap for global optimum is given by

( ) /Ggap UB GLB GLB= − . Due to the duality gap, this algorithm stops after a finite number

of iterations. As will be shown in the computational results, the global gaps are quite small.

7. Computational Results In order to illustrate the application of the proposed solution strategies, we present

computational experiments for five medium and large scale instances on an IBM T40 laptop

with Intel 1.50GHz CPU and 512 MB RAM. The proposed solution procedure is coded in

GAMS 22.8.1. The MILP problems are solved using CPLEX 11.0.1, the NLP problems in Step

2 of the Lagrangean solution approach are solved with solver CONOPT 3.3, and the global

optimizer used in the computational experiments is BARON 8.1.4.

7.1. Input Parameters

Since we consider large size problems, most of the input data are generated randomly. The

safety stock factors for DCs ( 1 jλ ) and customers ( 2kλ ) are the same and equal to 1.96, which

corresponds to 97.5% service level is demand is normally distrubted. We consider 365 days in

a year ( χ ). The guaranteed service time of the last echelon customer demand zones ( kR ) are

set to 0. The annual fixed costs ($/year) to install the DCs ( jf ) are generated uniformly on

U[150,000, 160,000] and the variable cost coefficient ( jg , $/ ton · year) are generated

uniformly on U[0.01, 0.1]. The guaranteed service times of the plants ( iSI , days) are set as

integers uniformly on U[1, 5]. The order processing time ( 1ijt , days) between plants and DCs

are generated as integers uniformly on U[1, 7], and the order processing time ( 2 jkt , days)

between DCs and customer demand zones are generated as integers uniformly on U[1, 3]. The

unit transportation cost from plants to DCs ( 1ijc , $/ton) and from DCs to customer demand

zones ( 2 jkc , $/ton) are set to

1 1 [0.05, 0.1]ij ijc t U= ×

2 2 [0.05, 0.1]jk jkc t U= ×

-36-

The expected demand ( iμ ,ton/day) is generated uniformly on U[75, 150] and its standard

deviation ( iσ , ton/day) is generated uniformly on U[0, 50]. The daily unit pipeline and safety

stock inventory holding costs ( 1 /ijθ χ , 2 /jkθ χ , 1 /jh χ and 2 /kh χ ) are generated

uniformly on U[0.1, 1].

In this instance, the annual fixed DC installation cost ( jf ) is $50,000/year for all the DCs.

The variable cost coefficient of installing a DC ( jg ) at all the candidate locations is $0.5/(ton ·

year). The safety stock factors for DCs ( 1jλ ) and wholesalers ( 2kλ ) are the same and equal to

1.96. We consider 365 days in a year ( χ ). The guaranteed service times of the three vendors

( iSI ) are 3 days, 3 days and 4 days, respectively. The pipeline inventory holding cost is $1/(ton

· day) for all the DCs ( 1 /ijθ χ ) and wholesalers ( 2 /jkθ χ ), and the safety stock holding cost is

$1.5/(ton · day) for all the DCs ( 1 /jh χ ) and wholesalers ( 2 /kh χ ).

The number of intervals P1, P2, P3 , which are required to approximate each of the

univariate concave terms jNZV , kL and jkL in (P3) and (APLPj), are all set to twenty.

All the breakpoints of the piecewise linear function are evenly distributed between the lower

and upper bound of the variables to be approximated. We perform the testing on 5 problem

instances of the model by varying the parameter settings.

7.2. Performance

The problem sizes for the five instances we considered in this example are given in Table

11. The numbers in i, j, k columns stand for the number of plants, potential DCs and customer

demand zones of each instance. The computational results of the five instances are given in

Table 12. We solve each instance with three solution approaches. The first approach is solving

the original MINLP problem (P0) directly with the global optimizer BARON. The second

approach is to first solve the initialization MILP problem (P2) with at most one hour

computational time, then use the best solution obtained from (P2) as the initial point to solve

the MINLP problem (P1) with DICOPT and SBB solvers. The third approach is the

Lagrangean relaxation algorithm as presented in Section 6.3.4. In the Lagrangean relaxation

algorithm, we solve the NLP relaxation of model (P1) to obtain the initial vector of Lagrangean

multipliers in Step 1, and solve the modified Lagrangean relaxation subproblem (APLRj) with

CONOPT 3.3 in Step 2. As we can see, global optimal solutions could not be obtained for 10

-37-

hours by solving these problems directly with BARON, or solving the reformulated problem

(P1) with the initialization scheme. However, optimal solutions within 1% global optimality

are obtained for the instances by using the proposed algorithm.

[Figure 14, (a), (b)] Furthermore, we can see from Table 12 that only the medium size instance with 2 plants,

20 potential DCs and 20 customer demand zones, can be solved directly with BARON to

obtain a feasible solution. For all the other instances, BARON fails to return any bounds or

feasible solutions due to the large size of these problems. By solving the reformulated problem

(P1) of the medium scale instance with convex MINLP solvers DICOPT and SBB after the

initialization scheme, we can see that a better feasible solution can be obtained with smaller

global optimality gap and much shorter computational times (140.3 and 163.6 CPU seconds,

respectively). However, the proposed Lagrangean relaxation and decomposition algorithm can

obtain much better solution ($1,776,969 vs. $1,889,577 and $1,820,174) within 0.06% of

global optimality in much shorter time 175.0 CPU seconds. For the other large scale instances,

BARON cannot finish preprocessing after 10 hours if solving the reformulated problem (P1).

In contrast the proposed Lagrangean relaxation algorithm can solve all the instances with 1%

optimality in less than 10 hours of computational time as shown in Table 12. Therefore, we can

see the significant advantage of using the proposed Lagrangean relaxation algorithm for

solving medium and large scale instances.

The change of bounds during the Lagrangean relaxation and decomposition algorithm for

a large scale instance with 3 plants, 50 potential DCs and 150 customer demand zones are

given in Figure 14. We can see that as iterations proceed, the upper bound, which is the

objective function value of a feasible solution, keeps decreasing, and both the incumbent lower

bound and the global lower bound keep increasing, until the termination criterion is satisfied.

We should note that the global optimality gap is given by the upper bound and the global lower

bound instead of the incumbent lower bound. The reason is that the incumbent lower bound

comes from the MILP subproblems (APLRj), which are not solved with BARON but CPLEX ,

but the global lower bounds come from the piecewise linear lower bounding MILP problem

(APLPj), which is solved to global optimal using CPLEX, and thus provides a valid lower

bound of the global optimal objective function value of the full problem.

[Table 11] [Table 12]

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8. Conclusion In this paper, we present an MINLP model that determines the optimal network structure,

transportation and inventory levels of a multi-echelon supply chain with the presence of

customer demand uncertainty. The well-known guaranteed service approach is used to model

the multi-echelon inventory system. The risk pooling effect is also taken into account in the

model by consolidating the demands in the downstream nodes to their upstream nodes.

Examples on supply chains for industrial gases and performance chemicals are presented to

illustrate the applicability of the proposed model. To solve the resulting MINLP problem

efficiently for large scale instances, a decomposition algorithm, based on Lagrangean

relaxation and piecewise linear approximation was proposed. Computational experiments on

large scale problems show that the proposed algorithm can obtain global or near-global optimal

solutions (typically within 1% of the global optimum) in modest computational expense

without the need of a global optimizer.

This research can be extended to address the responsiveness issue53, 54 of supply chains and

will be given in a further paper.

Acknowledgment The authors acknowledge the financial supports from the National Science Foundation

under Grant No. DMI-0556090 and No. OCI- 0750826, and Pennsylvania Infrastructure

Technology Alliance (PITA). Fengqi You is grateful to Prof. S. Willems (Boston University)

and Prof. S. C. Graves (MIT) for helpful discussions.

Appendix:

Proposition 1. If ( * * * * * *, , , , ,ij j jk j j kX Y Z S N L ) is a feasible solution of problem (P1) with objective

function value *W , then ( * * * * * *, , , , ,ij j jk j j jkX Y Z S N L ), where * 0jkL = if * 0jkZ = and * *jk kL L= if

* 1jkZ = , is a feasible solution of problem (AP) and the associated objective function value of

(AP) is *W . If ( * * * * * *, , , , ,ij j jk j j jkX Y Z S N L ) is a feasible solution of problem (AP) with objective

function value *W , then ( * * * * * *, , , , ,ij j jk j j kX Y Z S N L ), where * *k jkj J

L L∈

=∑ , is a feasible solution

of problem (P1) and the associated objective function value of (P1) is *W

Proof:

-39-

If ( * * * * * *, , , , ,ij j jk j j kX Y Z S N L ) is a feasible solution of problem (P1), it is easy to see that this

solution satisfies all the constraints of (AP) except (39) and (40). Constraints (39) and (40)

imply that the variable 0jkL = if 0jkZ = , and if 1jkZ = , and that we should have

jk j jk kL S t R≥ + − , which is the equivalent to constraint (32) for 1jkZ = . Thus, by setting

* 0jkL = if * 0jkZ = and * *jk kL L= if * 1jkZ = , we have the solution ( * * * * * *, , , , ,ij j jk j j jkX Y Z S N L ) that

satisfies all the constraints of (AP), and thus it is a feasible solution of (AP).

Because * 0jkL = if * 0jkZ = and * *jk kL L= if * 1jkZ = , and the following constraint

included in both (P1) and (AP),

1jkj J

Z∈

=∑ , k∀ (10)

which means for each k, there is only one * 1jkZ = . So we can have

* *k jk

j J

L L∈

= ∑ , k∀ (A1)

Because *W is the objective function value of (P1) corresponding to the solution

( * * * * * *, , , , ,ij j jk j j kX Y Z S N L ), we have

* * * * * *1 2j j ijk ijk jk jk j j k kj J i I j J k K j J k K j J k K

W f Y A XZ B Z q NZV q L∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

= + + + +∑ ∑∑∑ ∑∑ ∑ ∑ (A2)

Given (A1) and (A2), we can have

* * * * * *1 2j j ijk ijk jk jk j j k jkj J i I j J k K j J k K j J j J k K

W f Y A XZ B Z q NZV q L∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈

= + + + +∑ ∑∑∑ ∑∑ ∑ ∑∑ (A3)

which is the objective function value of (AP) corresponding to the solution

( * * * * * *, , , , ,ij j jk j j jkX Y Z S N L ).

Similarly, it is easy to show that if ( * * * * * *, , , , ,ij j jk j j jkX Y Z S N L ) is a solution of (AP), by

setting * *k jkj J

L L∈

= ∑ , we can have ( * * * * * *, , , , ,ij j jk j j kX Y Z S N L ) satisfying all the constraints of

(P1), and thus it is a feasible solution of (P1). On the other hand, given (A1) and (A3), we can

easily derive (A2).

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River, NJ, 2003.

(2) Grossmann, I. E., Enterprise-wide Optimization: A New Frontier in Process Systems Engineering. AIChE

Journal 2005, 51, 1846.

(3) Zipkin, P. H., Foundations of Inventory Management. McGraw-Hill: Boston, MA, 2000.

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(4) Simpson, K. F., In-process inventories. Operations Research 1958, 6, 863.

(5) Clark, A.; Scarf, S., Optimal policies for a multi-echelon inventory problem. Management Science 1960, 6,

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List of Figure Captions Figure 1 Periodical review base stock inventory policy

Figure 1 (a) Review period greater than replenishment lead time

Figure 1 (b) Review period less than replenishment lead time

Figure 2 Safety stock level for normally distributed demand

Figure 3 Timing Relationship in guaranteed service approach

Figure 4 Times in guaranteed service approach for in a three-stage serial inventory system

Figure 5 Supply chain network structure (three echelons)

Figure 6 Secant of a univariate square root term

Figure 7 LOX supply chain network superstructure for Example A

Figure 8 Optimal network structure for the LOX supply chain in Example A

Figure 8 (a) Without considering inventory costs, total cost = $108,329

Figure 8 (b) Considering inventory costs, total cost = $152,107

Figure 9 Cost component of the LOX supply chain in Example A

Figure 10 Acetic acid supply chain network superstructure for Example B

Figure 11 Optimal network structure for the acetic acid supply chain (Example B)

Figure 12 Cost component of the acetic acid supply chain in Example B

Figure 13 Piecewise linear function to approximate square root term

Figure 14 Bounds of each iteration of the Lagrangean relaxation algorithm for instance with

3 plants, 50 potential DCs and 150 customer demand zones

Figure 14 (a) Bounds for Iteration 1 to Iteration 33 (the last iteration)

Figure 14 (b) Bounds for Iteration 13 to Iteration 33 (the last iteration)

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Time

Inve

ntor

y

Lead time l

Place order

Receive order

Safety Stock Level

Base-Stock Level S

Review period p

Inventory position

Inventory on hand

a) Review period greater than replenishment lead time

Time

Inve

ntor

y

Lead time l

Place order

Receive order

Safety Stock Level

Base-Stock Level S

Review period p

Inventory position

Inventory on hand

b) Review period less than replenishment lead time

Figure 1 Periodical review base stock inventory policy

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Safety Stock Figure 2 Safety stock level for normally distributed demand

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tjTj-1

Tj

NLTj

Figure 3 Timing Relationship in guaranteed service approach

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1 2 3

t1 t2 t3

T1 T2 T3

Figure 4 Times in guaranteed service approach in a three-stage serial inventory

system

-48-

Figure 5 Supply chain network structure (three echelons)

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secant

x

U

xx

x

y

Figure 6 Secant of a univariate square root term

-50-

Figure 7 LOX supply chain network superstructure for Example A

-51-

(a) Without considering inventory costs, total cost = $108,329

(b) Considering inventory costs, total cost = $152,107

Figure 8 Optimal network structure for the LOX supply chain in Example A

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$-

$10,000

$20,000

$30,000

$40,000

$50,000

$60,000

$70,000

$80,000

$90,000

DC installation Transportation Pipeline inventory Safety stocks

without Inventory Costswith Inventory Costs

Figure 9 Cost component of the LOX supply chain in Example A

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Figure 10 Acetic acid supply chain network superstructure for Example B

-54-

Figure 11 Optimal network structure for the acetic acid supply chain (Example B)

-55-

21%

37%

37%

5%

DC installationTransportationPipeline inventorySafety stocks

Figure 12 Cost component of the acetic acid supply chain in Example B

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M2M1 M3 M4

F2

F3

F1

C1

C2

C3

Figure 13 Piecewise linear function to approximate square root term

-57-

-10000000

-5000000

0

5000000

10000000

15000000

20000000

25000000

30000000

35000000

0 5 10 15 20 25 30

Iterations

Incumbent Lower BoundGlobal Lower BoundUpper Bound

(a) Bounds for Iteration 1 to Iteration 33 (the last iteration)

12180000

12200000

12220000

12240000

12260000

12280000

12300000

12 17 22 27 32Iterations

Incumbent Lower BoundGlobal Lower BoundUpper Bound

(b) Bounds for Iteration 13 to Iteration 33 (the last iteration)

Figure 14 Bounds of each iteration of the Lagrangean relaxation algorithm for instance with 3 plants, 50 potential DCs and 150 customer demand zones

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Table 1 Parameters for demand uncertainty for Example A Mean demand iμ (liters/day) Standard Deviation iσ (liters/day)

Customer 1 257 150 Customer 2 86 25 Customer 3 194 120 Customer 4 75 45 Customer 5 292 64 Customer 6 95 30

Table 2 Order processing time ( 1ijt ) between plants and DCs (days) for Example A

DC1 DC2 DC3 Plant 1 7 4 2 Plant 2 2 4 7

Table 3 Order processing time ( 2 jkt ) between DCs and customers (days)

Cust. 1 Cust. 2 Cust. 3 Cust. 4 Cust. 5 Cust. 6 DC1 2 2 3 3 4 4 DC2 4 4 1 1 4 4 DC3 4 4 3 3 2 2

Table 4 Unit transportation Cost ( 1ijc ) from plants to DCs ($/liter)

DC1 DC2 DC3 Plant 1 0.24 0.20 0.20 Plant 2 0.18 0.19 0.23

Table 5 Unit transportation Cost ( 2 jkc ) from DCs to customers ($/liter)

Cust. 1 Cust. 2 Cust. 3 Cust. 4 Cust. 5 Cust. 6 DC1 0.01 0.03 0.10 0.44 1.60 2.30 DC2 1.50 0.25 0.01 0.02 0.25 1.50 DC3 2.27 1.73 0.51 0.10 0.01 0.03

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Table 6 Parameters for demand uncertainty for Example B Mean demand iμ (ton/day) Standard Deviation iσ (ton/day)

Wholesaler 1 250 150 Wholesaler 2 180 75 Wholesaler 3 150 80 Wholesaler 4 160 45

Table 7 Order processing time ( 1ijt ) between vendors and DCs (days) for Example B

DC1 DC2 DC3 Vendor 1 4 4 2 Vendor 2 2 4 3 Vendor 3 3 4 4

Table 8 Order processing time ( 2 jkt ) between DCs and wholesalers (days)

Wholesaler 1 Wholesaler 2 Wholesaler 3 Wholesaler 4 DC1 2 2 3 3 DC2 4 4 1 1 DC3 4 4 3 3

Table 9 Unit transportation Cost ( 1ijc ) from vendors to DCs ($/ton)

DC1 DC2 DC3 Vendor 1 1.8 1.6 2.0 Vendor 2 2.4 2.2 1.3 Vendor 3 2.0 1.3 2.5

Table 10 Unit transportation Cost ( 2 jkc ) from DCs to wholesalers ($/ton)

Wholesaler 1 Wholesaler 2 Wholesaler 3 Wholesaler 4 DC1 1.0 3.3 4.0 7.4 DC2 1.0 0.5 0.1 2.0 DC3 7.7 7.3 5.1 0.1

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Table 11 Problem sizes for the medium and large scale instances MINLP (P0) MINLP (P1) and MILP (P2) MILP (APLPj), 20 intervals NLP (APLRj), X and Z fixed i j k Dis. Var. Con. Var. Const. Dis. Var. Con. Var. Const. Dis. Var. Con. Var. Const. Con. Var. Const. NL-NZ

2 20 20 460 60 480 460 2,480 5,300 442 564 1,165 144 282 20 5 30 50 1,680 110 1,660 1,680 13,640 33,190 1,105 2,658 3,355 504 1,152 50 10 50 100 5,550 200 5,300 5,550 70,250 185,350 2,210 5,814 8,205 1,504 3,802 100 20 50 100 6,050 200 5,300 6,050 120,250 335,350 2,220 6,824 11,205 2,504 6802 100 3 50 150 7,700 250 7,900 7,700 52,800 120,450 3,303 4,352 9,155 1,204 2,552 150

“Dis. Var.” = discrete variables; “Con. Var.” = continuos variable; “Const.” = constraints; “NL-NZ” = nonlinear nonzeros

Table 12 Comparison of the performance of the algorithms for medium and large scale instances * No solution or bounds were returned due to solver failure.

** No solution was returned after 10 hours

Solve (P0) directly with BARON Solve (P2) with CPLEX for at most 1 hour,then solve (P1) with DICOPT or SBB Lagrangean Relaxation Algorithm

DICOPT SBB i j k

Solution LB Gap Time (s) Solution Time(s) Solution Time(s)

Solution Global LB GlobalGap

Time (s) Iter.

2 20 20 1,889,577 1,159,841 62.92% 36,000 1,820,174 140.3 1,813,541 163.6 1,776,969 1,775,957 0.06% 175.0 11

5 30 50 ---* ---* ---* 36,000 ---** 36,000 ---** 36,000 4,417,353 4,403,582 0.31% 3,279 24

10 50 100 ---* ---* ---* 36,000 ---** 36,000 ---** 36,000 7,512,609 7,477,584 0.47% 27,719 42

20 50 100 ---* ---* ---* 36,000 ---** 36,000 ---** 36,000 5,620,045 5,576,126 0.79% 27,748 53

3 50 150 ---* ---* ---* 36,000 ---** 36,000 ---** 36,000 12,291,296 12,276,483 0.12% 16,112 32